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S97-11

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S97-11  [#permalink]

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New post 16 Sep 2014, 00:51
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A
B
C
D
E

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Question Stats:

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Set S consists of 5 values, not necessarily in ascending order: {4, 8, 12, 16, \(x\)}. For how many values of \(x\) does the mean of set S equal the median of set \(S\)?

A. Zero
B. One
C. Two
D. Three
E. More than three

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Re S97-11  [#permalink]

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New post 16 Sep 2014, 00:51
1
Official Solution:

Set S consists of 5 values, not necessarily in ascending order: {4, 8, 12, 16, \(x\)}. For how many values of \(x\) does the mean of set S equal the median of set \(S\)?

A. Zero
B. One
C. Two
D. Three
E. More than three


To solve this problem quickly, you might try to come up with likely values for \(x\) that would make the mean equal the median. One sort of set for which the mean equals the median is a set with values symmetrically spaced around its mean/median. The values do not have to be evenly spaced.

Three values that would make the set symmetrical are 0, 10, and 20:

{0, 4, 8, 12, 16}

{4, 8, 10, 12, 16}

{4, 8, 12, 16, 20}

We are down to choices (D) and (E). Now, can we prove that no other values of \(x\) make the mean equal the median? After all, some non-symmetrical sets have their mean equal to their median: for instance, {1, 1, 2, 2.5, 3.5}. All you need to do is make the "residuals," or differences, around the middle value cancel out (in the case above, the values to the left of 2 are 1 & 1, leaving a total residual of -2, while the values to the right of 2 are 2.5 and 3.5, leaving a total residual of +2).

Well, we can set up three scenarios, each with a relevant equation.

(1) If \(x\) is less than or equal to 8, then the median is equal to 8. We now set the mean equal to the median:
\(\frac{40 + x}{5} = 8\)
\(40 + x = 40\)
\(x = 0\)

(2) If \(x\) is between 8 and 12, then the median is equal to \(x\). Again, we set the mean equal to the median:
\(\frac{40 + x}{5} = x\)
\(40 + x = 5x\)
\(40 = 4x\)
\(x = 10\)

(3) If \(x\) is greater than 12, then the median is equal to 12. Again, we set the mean equal to the median:
\(\frac{40 + x}{5} = 12\)
\(40 + x = 60\)
\(x = 20\)

We have now exhausted all the possibilities for \(x\). In fact, we did not have to actually compute the values of \(x\) in each case; rather, we could have simply realized that each equation is linear in \(x\) and so would have exactly one solution. Since there are three scenarios, there are exactly three values of \(x\) that satisfy the constraint of making the mean and the median equal. Indeed, if we had started with this approach, we might have gotten to the answer more quickly.


Answer: D
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Re: S97-11  [#permalink]

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New post 28 Aug 2016, 02:50
Bunuel wrote:
Official Solution:

Set S consists of 5 values, not necessarily in ascending order: {4, 8, 12, 16, \(x\)}. For how many values of \(x\) does the mean of set S equal the median of set \(S\)?

A. Zero
B. One
C. Two
D. Three
E. More than three


To solve this problem quickly, you might try to come up with likely values for \(x\) that would make the mean equal the median. One sort of set for which the mean equals the median is a set with values symmetrically spaced around its mean/median. The values do not have to be evenly spaced.

Three values that would make the set symmetrical are 0, 10, and 20:

{0, 4, 8, 12, 16}

{4, 8, 10, 12, 16}

{4, 8, 12, 16, 20}

We are down to choices (D) and (E). Now, can we prove that no other values of \(x\) make the mean equal the median? After all, some non-symmetrical sets have their mean equal to their median: for instance, {1, 1, 2, 2.5, 3.5}. All you need to do is make the "residuals," or differences, around the middle value cancel out (in the case above, the values to the left of 2 are 1 & 1, leaving a total residual of -2, while the values to the right of 2 are 2.5 and 3.5, leaving a total residual of +2).

Well, we can set up three scenarios, each with a relevant equation.

(1) If \(x\) is less than or equal to 8, then the median is equal to 8. We now set the mean equal to the median:
\(\frac{40 + x}{5} = 8\)
\(40 + x = 40\)
\(x = 0\)

(2) If \(x\) is between 8 and 12, then the median is equal to \(x\). Again, we set the mean equal to the median:
\(\frac{40 + x}{5} = x\)
\(40 + x = 5x\)
\(40 = 4x\)
\(x = 10\)

(3) If \(x\) is greater than 12, then the median is equal to 12. Again, we set the mean equal to the median:
\(\frac{40 + x}{5} = 12\)
\(40 + x = 60\)
\(x = 20\)

We have now exhausted all the possibilities for \(x\). In fact, we did not have to actually compute the values of \(x\) in each case; rather, we could have simply realized that each equation is linear in \(x\) and so would have exactly one solution. Since there are three scenarios, there are exactly three values of \(x\) that satisfy the constraint of making the mean and the median equal. Indeed, if we had started with this approach, we might have gotten to the answer more quickly.


Answer: D







Hi,


Isnt it necessary for the values to be in AP,for the mean and median to be equal?
Math Expert
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Joined: 02 Sep 2009
Posts: 53066
Re: S97-11  [#permalink]

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New post 28 Aug 2016, 04:41
ashima09 wrote:
Bunuel wrote:
Official Solution:

Set S consists of 5 values, not necessarily in ascending order: {4, 8, 12, 16, \(x\)}. For how many values of \(x\) does the mean of set S equal the median of set \(S\)?

A. Zero
B. One
C. Two
D. Three
E. More than three


To solve this problem quickly, you might try to come up with likely values for \(x\) that would make the mean equal the median. One sort of set for which the mean equals the median is a set with values symmetrically spaced around its mean/median. The values do not have to be evenly spaced.

Three values that would make the set symmetrical are 0, 10, and 20:

{0, 4, 8, 12, 16}

{4, 8, 10, 12, 16}

{4, 8, 12, 16, 20}

We are down to choices (D) and (E). Now, can we prove that no other values of \(x\) make the mean equal the median? After all, some non-symmetrical sets have their mean equal to their median: for instance, {1, 1, 2, 2.5, 3.5}. All you need to do is make the "residuals," or differences, around the middle value cancel out (in the case above, the values to the left of 2 are 1 & 1, leaving a total residual of -2, while the values to the right of 2 are 2.5 and 3.5, leaving a total residual of +2).

Well, we can set up three scenarios, each with a relevant equation.

(1) If \(x\) is less than or equal to 8, then the median is equal to 8. We now set the mean equal to the median:
\(\frac{40 + x}{5} = 8\)
\(40 + x = 40\)
\(x = 0\)

(2) If \(x\) is between 8 and 12, then the median is equal to \(x\). Again, we set the mean equal to the median:
\(\frac{40 + x}{5} = x\)
\(40 + x = 5x\)
\(40 = 4x\)
\(x = 10\)

(3) If \(x\) is greater than 12, then the median is equal to 12. Again, we set the mean equal to the median:
\(\frac{40 + x}{5} = 12\)
\(40 + x = 60\)
\(x = 20\)

We have now exhausted all the possibilities for \(x\). In fact, we did not have to actually compute the values of \(x\) in each case; rather, we could have simply realized that each equation is linear in \(x\) and so would have exactly one solution. Since there are three scenarios, there are exactly three values of \(x\) that satisfy the constraint of making the mean and the median equal. Indeed, if we had started with this approach, we might have gotten to the answer more quickly.


Answer: D







Hi,


Isnt it necessary for the values to be in AP,for the mean and median to be equal?

___________________
No, consider {1, 1, 2, 2}.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: S97-11  [#permalink]

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New post 19 Apr 2018, 12:34
1
Hi,
It does not say anywhere, that the value of x should be positive or equal to zero.
This bring a confusion and reflects to the answer.


Kind regards
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Joined: 02 Sep 2009
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Re: S97-11  [#permalink]

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New post 20 Apr 2018, 01:17
StudyHard11 wrote:
Hi,
It does not say anywhere, that the value of x should be positive or equal to zero.
This bring a confusion and reflects to the answer.


Kind regards


The question asks about all possible values of x, negative, 0, positive, for which he mean of set S will be equal the median of set S. There are only three values possible: 0, 10 and 20.
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New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: S97-11  [#permalink]

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New post 24 Jun 2018, 16:55
Bunuel wrote:
Official Solution:

Set S consists of 5 values, not necessarily in ascending order: {4, 8, 12, 16, \(x\)}. For how many values of \(x\) does the mean of set S equal the median of set \(S\)?

A. Zero
B. One
C. Two
D. Three
E. More than three


To solve this problem quickly, you might try to come up with likely values for \(x\) that would make the mean equal the median. One sort of set for which the mean equals the median is a set with values symmetrically spaced around its mean/median. The values do not have to be evenly spaced.

Three values that would make the set symmetrical are 0, 10, and 20:

{0, 4, 8, 12, 16}

{4, 8, 10, 12, 16}

{4, 8, 12, 16, 20}

We are down to choices (D) and (E). Now, can we prove that no other values of \(x\) make the mean equal the median? After all, some non-symmetrical sets have their mean equal to their median: for instance, {1, 1, 2, 2.5, 3.5}. All you need to do is make the "residuals," or differences, around the middle value cancel out (in the case above, the values to the left of 2 are 1 & 1, leaving a total residual of -2, while the values to the right of 2 are 2.5 and 3.5, leaving a total residual of +2).

Well, we can set up three scenarios, each with a relevant equation.

(1) If \(x\) is less than or equal to 8, then the median is equal to 8. We now set the mean equal to the median:
\(\frac{40 + x}{5} = 8\)
\(40 + x = 40\)
\(x = 0\)

(2) If \(x\) is between 8 and 12, then the median is equal to \(x\). Again, we set the mean equal to the median:
\(\frac{40 + x}{5} = x\)
\(40 + x = 5x\)
\(40 = 4x\)
\(x = 10\)

(3) If \(x\) is greater than 12, then the median is equal to 12. Again, we set the mean equal to the median:
\(\frac{40 + x}{5} = 12\)
\(40 + x = 60\)
\(x = 20\)

We have now exhausted all the possibilities for \(x\). In fact, we did not have to actually compute the values of \(x\) in each case; rather, we could have simply realized that each equation is linear in \(x\) and so would have exactly one solution. Since there are three scenarios, there are exactly three values of \(x\) that satisfy the constraint of making the mean and the median equal. Indeed, if we had started with this approach, we might have gotten to the answer more quickly.


Answer: D


Hello,
It says that the numbers need not be ascending order. So, why aren't we taking the case {4, 8, 16, 12, x} ?
which implies x= 40. Therefore, values of x is more than three.
Thanks in advance!
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Re: S97-11  [#permalink]

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New post 12 Oct 2018, 08:09
ladhaa wrote:
Bunuel wrote:
Official Solution:

Set S consists of 5 values, not necessarily in ascending order: {4, 8, 12, 16, \(x\)}. For how many values of \(x\) does the mean of set S equal the median of set \(S\)?

A. Zero
B. One
C. Two
D. Three
E. More than three


To solve this problem quickly, you might try to come up with likely values for \(x\) that would make the mean equal the median. One sort of set for which the mean equals the median is a set with values symmetrically spaced around its mean/median. The values do not have to be evenly spaced.

Three values that would make the set symmetrical are 0, 10, and 20:

{0, 4, 8, 12, 16}

{4, 8, 10, 12, 16}

{4, 8, 12, 16, 20}

We are down to choices (D) and (E). Now, can we prove that no other values of \(x\) make the mean equal the median? After all, some non-symmetrical sets have their mean equal to their median: for instance, {1, 1, 2, 2.5, 3.5}. All you need to do is make the "residuals," or differences, around the middle value cancel out (in the case above, the values to the left of 2 are 1 & 1, leaving a total residual of -2, while the values to the right of 2 are 2.5 and 3.5, leaving a total residual of +2).

Well, we can set up three scenarios, each with a relevant equation.

(1) If \(x\) is less than or equal to 8, then the median is equal to 8. We now set the mean equal to the median:
\(\frac{40 + x}{5} = 8\)
\(40 + x = 40\)
\(x = 0\)

(2) If \(x\) is between 8 and 12, then the median is equal to \(x\). Again, we set the mean equal to the median:
\(\frac{40 + x}{5} = x\)
\(40 + x = 5x\)
\(40 = 4x\)
\(x = 10\)

(3) If \(x\) is greater than 12, then the median is equal to 12. Again, we set the mean equal to the median:
\(\frac{40 + x}{5} = 12\)
\(40 + x = 60\)
\(x = 20\)

We have now exhausted all the possibilities for \(x\). In fact, we did not have to actually compute the values of \(x\) in each case; rather, we could have simply realized that each equation is linear in \(x\) and so would have exactly one solution. Since there are three scenarios, there are exactly three values of \(x\) that satisfy the constraint of making the mean and the median equal. Indeed, if we had started with this approach, we might have gotten to the answer more quickly.


Answer: D


Hello,
It says that the numbers need not be ascending order. So, why aren't we taking the case {4, 8, 16, 12, x} ?
which implies x= 40. Therefore, values of x is more than three.
Thanks in advance!


In order to compute the median, values have to be in order (either ascending or descending). Therefore your statement is fallacious.
GMAT Club Bot
Re: S97-11   [#permalink] 12 Oct 2018, 08:09
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